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Papers by Valentin D. Burcea

arXiv (Cornell University), Aug 1, 2016

It is studied the convergence for Formal Holomorphic Mappings defined between Special Classes of ... more It is studied the convergence for Formal Holomorphic Mappings defined between Special Classes of C.-R. Singular Real-Analytic Submanifolds in Complex Spaces. In particular, it is solved the local Equivalence Problem in some particular cases.

arXiv (Cornell University), Jul 31, 2017

It is studied the Classification Problem for Formal (Holomorphic) Embeddings between Shilov Bound... more It is studied the Classification Problem for Formal (Holomorphic) Embeddings between Shilov Boundaries of Bounded Symmetric Domains of First Type situated in Complex Spaces of Different Dimensions. There are obtained two Classes of Equivalence.

Complex Variables and Elliptic Equations, 2016

Let (z, w) be the coordinates in C 2. We construct a normal form for a class of real formal surfa... more Let (z, w) be the coordinates in C 2. We construct a normal form for a class of real formal surfaces M ⊂ C 2 defined near a degenerate CR singularity p = 0 as follows w = P (z, z) + O |z| k 0 +1 , where P (z, z) is a real-valued homogeneous polynomial in (z, z) of degree k 0 ≥ 3 such that the coefficients of z k 0 and z k 0 are vanishing.

Methods and Applications of Analysis, 2013

We construct a family of analytic discs attached to a real submanifold M ⊂ C N+1 of codimension 2... more We construct a family of analytic discs attached to a real submanifold M ⊂ C N+1 of codimension 2 near a CR singularity. These discs are mutually disjoint and form a smooth hypersurface M with boundary M in a neighborhood of the CR singularity. As an application we prove that if p is a flat-elliptic CR singularity and if M is nowhere minimal at its CR points and does not contain a complex manifold of dimension (n − 2), then M is a smooth Levi-flat hypersurface. Moreover, if M is real analytic we obtain that M is real-analytic across the boundary manifold M .

Advances in Mathematics, 2013

We construct a complete formal normal form for a real 2-codimensional submanifold M ⊂ C N +1 near... more We construct a complete formal normal form for a real 2-codimensional submanifold M ⊂ C N +1 near a CR singularity approximating the sphere. This result gives a higher dimensional extension of Huang-Yin normal form in C 2 .

There are solved several related problems to Formal (Holomorphic) Segre preserving Mappings of a ... more There are solved several related problems to Formal (Holomorphic) Segre preserving Mappings of a (large) Class of Real-Formal Hypersurfaces in ℂ^2.

Abstract. It is studied the local Equivalence Problem in Complex Analysis. It is proven that the ... more Abstract. It is studied the local Equivalence Problem in Complex Analysis. It is proven that the Formal (Holomorphic Segre Preserving) Equivalences, of Real-Formal Hypersurfaces in C, are determined by their 1-jets. These Equivalences are also inversable and their inverses are also smooth. Then, it is concluded that any Formal Holomorphic Segre Preserving Mapping, between the complexifications of two RealAnalytic Hypersurfaces or Real-Algebraic Hypersurfaces in C, is convergent or algebraic. In particular, there are derived formal constructions of normal form type, which are convergent if the source manifold is analytic,and respectively algebraic if the source manifold is algebraic.

arXiv (Cornell University), Aug 1, 2016

It is studied the convergence for Formal Holomorphic Mappings defined between Special Classes of ... more It is studied the convergence for Formal Holomorphic Mappings defined between Special Classes of C.-R. Singular Real-Analytic Submanifolds in Complex Spaces. In particular, it is solved the local Equivalence Problem in some particular cases.

arXiv (Cornell University), Jul 31, 2017

It is studied the Classification Problem for Formal (Holomorphic) Embeddings between Shilov Bound... more It is studied the Classification Problem for Formal (Holomorphic) Embeddings between Shilov Boundaries of Bounded Symmetric Domains of First Type situated in Complex Spaces of Different Dimensions. There are obtained two Classes of Equivalence.

Complex Variables and Elliptic Equations, 2016

Let (z, w) be the coordinates in C 2. We construct a normal form for a class of real formal surfa... more Let (z, w) be the coordinates in C 2. We construct a normal form for a class of real formal surfaces M ⊂ C 2 defined near a degenerate CR singularity p = 0 as follows w = P (z, z) + O |z| k 0 +1 , where P (z, z) is a real-valued homogeneous polynomial in (z, z) of degree k 0 ≥ 3 such that the coefficients of z k 0 and z k 0 are vanishing.

Methods and Applications of Analysis, 2013

We construct a family of analytic discs attached to a real submanifold M ⊂ C N+1 of codimension 2... more We construct a family of analytic discs attached to a real submanifold M ⊂ C N+1 of codimension 2 near a CR singularity. These discs are mutually disjoint and form a smooth hypersurface M with boundary M in a neighborhood of the CR singularity. As an application we prove that if p is a flat-elliptic CR singularity and if M is nowhere minimal at its CR points and does not contain a complex manifold of dimension (n − 2), then M is a smooth Levi-flat hypersurface. Moreover, if M is real analytic we obtain that M is real-analytic across the boundary manifold M .

Advances in Mathematics, 2013

We construct a complete formal normal form for a real 2-codimensional submanifold M ⊂ C N +1 near... more We construct a complete formal normal form for a real 2-codimensional submanifold M ⊂ C N +1 near a CR singularity approximating the sphere. This result gives a higher dimensional extension of Huang-Yin normal form in C 2 .

There are solved several related problems to Formal (Holomorphic) Segre preserving Mappings of a ... more There are solved several related problems to Formal (Holomorphic) Segre preserving Mappings of a (large) Class of Real-Formal Hypersurfaces in ℂ^2.

Abstract. It is studied the local Equivalence Problem in Complex Analysis. It is proven that the ... more Abstract. It is studied the local Equivalence Problem in Complex Analysis. It is proven that the Formal (Holomorphic Segre Preserving) Equivalences, of Real-Formal Hypersurfaces in C, are determined by their 1-jets. These Equivalences are also inversable and their inverses are also smooth. Then, it is concluded that any Formal Holomorphic Segre Preserving Mapping, between the complexifications of two RealAnalytic Hypersurfaces or Real-Algebraic Hypersurfaces in C, is convergent or algebraic. In particular, there are derived formal constructions of normal form type, which are convergent if the source manifold is analytic,and respectively algebraic if the source manifold is algebraic.